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Finite quantitative soundness of the CNA test with side conditions

Lax323828.CNAQuantitativeSoundness · concepts/Lax323828/CNAQuantitativeSoundness.lean · lax-323828

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    Natural Language Statement

    Theorem

    Combine soundness at each evaluation point with concentration of random label fibers. Every accepted run agrees with an entire fiber. Unless a fiber is unusually small, averaging over the possible evaluation points costs only a factor twice the number of labels, independently of the number of words. Projection makes the decoding set independent of the side condition.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    1import Lax323828.CNAPointSoundness
    2import Lax323828.CNASoundness
    3import Lax323828.RandomFibers
    4
    5/-!
    6---
    7title: Finite quantitative soundness of the CNA test with side conditions
    8type: theorem
    9---
    10Combine soundness at each evaluation point with concentration of random
    11label fibers. Every accepted run agrees with an entire fiber. Unless a
    12fiber is unusually small, averaging over the possible evaluation points
    13costs only a factor twice the number of labels, independently of the
    14number of words. Projection makes the decoding set independent of the
    15side condition.
    16-/
    17
    18namespace Lax323828.CNAQuantitativeSoundness
    19
    20open LongCode BooleanFourier CNAPointSoundness CNASoundness FiniteProbability
    21
    22noncomputable def totalBound (l m r s w : ℕ) (τ q : ℝ) : ℝ :=
    23 (2 : ℝ) ^ s * Real.exp (-((2 : ℝ) ^ w) / (8 * ((2 : ℝ) ^ s) ^ 2)) +
    24 2 * (2 : ℝ) ^ s * pointBound l m r (2 ^ s) τ q
    25
    26axiom quantitative_soundness (w s l m r : ℕ) (hs : 0 < s)
    27 (hl : 0 < l) (hlN : l < 2 ^ s)
    28 (τ q : ℝ) (hτ : 0 < τ) (hq : 0 ≤ q) (hq1 : q ≤ 1)
    29 (hr : 2 * r ≤ m) (hm : Even m)
    30 (hlarge : (l : ℝ) / ((2 : ℝ) ^ s - l) / τ ≤ 1 / 3)
    31 (A : Table w) (h : Coordinate w) :
    32 probability (BadWithCondition (s := s) A (decoding (fun g ↦ sign (A g)) l τ) h) ≤
    33 totalBound l m r s w τ q
    34
    35end Lax323828.CNAQuantitativeSoundness
    36
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